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The Arf-Kervaire invariant problem in algebraic topology: Introduction. (English) Zbl 1223.55009
Jerison, David (ed.) et al., Current developments in mathematics, 2009. Somerville, MA: International Press (ISBN 978-1-57146-146-9/hbk). 23-58 (2010).
The authors have proved the non-existence of elements of Kervaire invariant one, which is the solution to one of the oldest problems in algebraic topology; cf. their preprint [On the non-existence of elements of Kervaire invariant one, arXiv:0908.3724]. As the authors write, this paper gives the history and background of the problem, along with a short summary of their solution to it and a description of some of the tools they use. Indeed, they summarize them effectively. This will be a great help for those who have interest in the problem or are going to understand their proof.
For the entire collection see [Zbl 1205.00075].

##### MSC:
 55Q91 Equivariant homotopy groups 55Q45 Stable homotopy of spheres 57R15 Specialized structures on manifolds (spin manifolds, framed manifolds, etc.) 55P42 Stable homotopy theory, spectra 55-02 Research exposition (monographs, survey articles) pertaining to algebraic topology 55P91 Equivariant homotopy theory in algebraic topology 57R55 Differentiable structures in differential topology 57R60 Homotopy spheres, Poincaré conjecture 57R77 Complex cobordism ($$\mathrm{U}$$- and $$\mathrm{SU}$$-cobordism) 57R85 Equivariant cobordism
##### Keywords:
Kervaire invariant; Arf invariant; homotopy